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Consider the following two metric spaces in $\mathbb{R}^n$:
$$
d_2(\mathbf{x},\mathbf{y}) = \sum_{n = 1}^{n}|x_i - y_i|.
$$
Prove that the ball $B(\mathbf{a},r)$ has the geometric appearance indicated:
In $(\mathbb{R}^2,d_2)$, a square with diagonals parallel to the axes.
Take a ball $B((x,y),r)$ for $r > 0$. Let $(a,b)\in B((x,y),r)$.
By the metric, we have $||x-a| + |y-b|| < r$
Now what?
$$
d_2(\mathbf{x},\mathbf{y}) = \sum_{n = 1}^{n}|x_i - y_i|.
$$
Prove that the ball $B(\mathbf{a},r)$ has the geometric appearance indicated:
In $(\mathbb{R}^2,d_2)$, a square with diagonals parallel to the axes.
Take a ball $B((x,y),r)$ for $r > 0$. Let $(a,b)\in B((x,y),r)$.
By the metric, we have $||x-a| + |y-b|| < r$
Now what?